The rule is unclear in this scenario if the "captive" heroes generate influence to an influence pool for use by their owner. It is not worded like the "prisoner" is in the core box set scenario Escape from Dol Guldur, which fully described the restrictions. The captives are still considered in play and nothing says that their influence would not be generated or unusable by the owner. Nor have I found anything in the FAQ or in this forum that I could uncover.
So, in playing the scenario... I used the captives influence pool to play cards which felt like the hobbits were still working against the Urik-Hai as in the book. They were unable to do anything else... But could be a target of the Burden - Overcome by Grief... Nothing says it would not be feasible.
Does anyone have another interpretation on how this should be played? I cannot believe the designers left this so open ended. Their running list of FAQ's grows ever longer as they obviously overlook describing fundamental aspects all the time. I wish they would bring in playtesters who are not such fanboys of the product to accurately polish up the text before printing.